None of the examples treat it that way. The first example clearly shows the Slow trait ending the creature's Action Phase. The second involves gaining the Slow trait after declaring an additional action. The third involves losing the Slow trait before it can take effect. None of these are inconsistent with Slow ending the Action Phase.
The FAQ describes the method for applying Slow in a way that can never happen if we accept the Codex literally.
1. We are told that the rules check to see if a creature can take actions when that action is declared.
2. The FAQ says that Slow can therefore prevent a creature from taking an action as long as the trait is applied before the creature declares.
3. If Slow really ended a creature's activation without an opportunity to reveal a Cheetah Speed, Slow would never trigger the rules test the FAQ describes.
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Let's take the examples one at a time.
Example: A creature begins its Action Phase by taking a move action to move into another zone. After it moves into the new zone, the opponent reveals an Enfeeble enchantment on the creature, which gives it the Slow trait. The creature was planning to make a quick melee attack on an enemy creature in the zone it just entered. However, because it acquired the Slow trait before it makes the quick action melee attack, it can no longer take that action and must end its Action Phase instead.
This example says that 1. The creature has Slow
THEREFORE
2a. The creature cannot attack and
2b. The creature "must end its Action Phase instead"
You're arguing a different thing
1. The creature has Slow
THEREFORE
2. The creature's Action Phase ends
THEREFORE
3. The creature cannot attack
Example: The same creature above enters the new zone and the opponent does not reveal Enfeeble on it, yet. It then declares a quick action melee attack against an enemy creature in that zone. During that melee attack, the opponent reveals the Enfeeble enchantment on the creature, giving it the Slow trait. Since it acquired the Slow trait after it has already started its attack action, the Slow trait does not affect or prevent that attack from continuing.
This is again written in terms of "preventing that attack" and not in terms of "ending that creature's action phase".
Example: A zone enchantment gives all creatures in that zone the Slow trait. A creature begins its Action Phase in that zone by taking a move action to move out of the zone. It has the Slow trait as it moves out of the zone, but as soon as it completes its move action it no longer is Slow. After moving out of the zone it may now take a quick action.
I've actually got a number of notes on this example, some of which may be relevant
1. If Slow did, in fact, end a creature's Action Phase as part of declaring the first move action (which I don't think either of us are arguing for, right?) this example would be wrong, because the creature's Action Phase would be over when it finished moving, and it wouldn't matter if it was Slow or not at that point.
2. Under the current rules, this example does actually have a small error: the creature loses Slow during the "Move To a New Zone" step, which is before the final "Entering Zone Triggers" step, and well before the end of the movement action. It's not quite the case that "as soon as it completes its move action it no longer is Slow." That actually happens during the move action. I suppose it's technically true, in the sense that "as soon as it's Sunday, Friday is over" would be true, but it's a bit misleading.
3. This example, written as it is, says that Slow is checked after completing the move action. But it doesn't say exactly when that check happens.
Usually, checks happen at the start of actions or phases, and we have reason to believe that Slow is no exception. The FAQ says, in the first sentence of the entry, "A creature checks whether or not it can take an additional action at the moment it would declare the action." To me that sounds like "Slow is checked by the rules when a creature tries to declare an action."
This would be in keeping with the precedent set by Necropian Vampiress using its ability to Fly and avoid gaining Hindered when it takes its move action.
I propose that the order of play is as follows:1. Move Action completes.
2. Enfeeble is revealed after the Move Action. The Creature gains Slow.
3. Cheetah Speed is revealed after the Move Action. The Creature gains Fast, and therefore loses both Fast and Slow. No further enchantments are revealed.
4. Attack Action is declared.
5. Action legality is checked, and since the creature doesn't have Slow, everything continues as normal.
The important key to this proposal is that Slow isn't checked when the Enfeeble is revealed, but only when the creature tries to act. Much the same way that you can reveal Rhino Hide whenever you want, but Armor +2 is only checked when you get to the Apply Damage step.
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I will add that I would strongly support your suggested change to make slow merely prevent taking additional actions, checked at the time you would declare said action. But until such a change is made, that is not how it currently works.
I try hard to separate my advocacy for errata from my judgement of What The Rules Are. But you're correct that I would like the rules to be as I describe them, in addition to thinking that they are that way already.